Risk, Uncertainty and Monetary Policy*
Geert Bekaert
Marie Hoerova
Marco Lo Duca
Columbia GSB
ECB
ECB
April 2010
Abstract
We document a strong co-movement between the VIX, the stock market option-based
implied volatility, and monetary policy. We decompose the VIX into two components, a
proxy for risk aversion and expected stock market volatility (“uncertainty”), and analyze
their dynamic interactions with monetary policy in a structural vector autoregressive
framework. A lax monetary policy decreases risk aversion after about six months.
Monetary authorities react to periods of high uncertainty by easing monetary policy.
These results are robust to controlling for business cycle movements. We further
investigate channels through which monetary policy may affect risk aversion, e.g.,
through its effects on broad liquidity measures and credit.
JEL Classification: E44; E 52; G12; G20; E32
Keywords: Monetary policy; Option implied volatility; Risk aversion; Uncertainty;
Business cycle; Stock market volatility dynamics
* We thank Gianni Amisano and Frank Smets for helpful comments and suggestions. Falk Bräuning and
Francesca Fabbri provided excellent research assistance. The views expressed do not necessarily reflect
those of the European Central Bank or the Eurosystem.
Electronic copy available at: http://ssrn.com/abstract=1561171
I. Introduction
A popular indicator of risk aversion in financial markets, the VIX index, shows strong comovements with measures of the monetary policy stance. Figure 1 considers the crosscorrelogram between the real interest rate (the Fed funds rate minus inflation), a measure
of the monetary policy stance, and the logarithm of end-of-month readings of the VIX
index. The VIX contract, traded on the Chicago Board Options Exchange, essentially
prices the “risk-neutral” expected stock market variance for the US S&P500 contract.
The correlogram reveals a very strong positive correlation between real interest rates and
future VIX levels. While the current VIX is positively associated with future real rates,
the relationship turns negative and significant after 13 months: high VIX readings are
correlated with expansionary monetary policy in the medium-run future.
The strong interaction between a “fear index” (Whaley (2000)) in the asset markets
and monetary policy indicators may have important implications for a number of
literatures. First, the recent crisis has rekindled the idea that lax monetary policy can be
conducive to financial instability. The Federal Reserve’s pattern of providing liquidity to
financial markets following market tensions, which became known as the “Greenspan
put”, has been cited as one of the contributing factors to the build up of a speculative
bubble prior to the 2007-09 financial crisis. 1 Whereas some rather informal stories have
linked monetary policy to risk-taking in financial markets (Rajan (2006), Adrian and Shin
(2008), Borio and Zhu (2008)), it is fair to say that no extant research establishes a firm
empirical link between monetary policy and risk aversion in asset markets. 2
Second, Bloom (2009) and Bloom, Floetotto and Jaimovich (2009) show that
heightened “economic uncertainty” decreases employment and output. It is therefore
conceivable that the monetary authority responds to uncertainty shocks, in order to affect
economic outcomes. However, the VIX index, used by Bloom (2009) to measure
uncertainty, can be decomposed into a component that reflects actual expected stock
1
Investors increasingly believed that when market conditions were to deteriorate, the Fed would step in and
inject liquidity until the outlook improved. The perception may have become embedded in asset pricing in
the form of higher valuations, narrower credit spreads, and excessive risk-taking. See, for example,
“Greenspan Put may be Encouraging Complacency,” Financial Times, December 8, 2000.
2
For recent empirical evidence that monetary policy affects the riskiness of loans granted by banks see, for
example, Altunbas, Gambacorta and Marquéz-Ibañez (2009), Ioannidou, Ongena and Peydró (2009),
Jiménez, Ongena, Peydró and Saurina (2009), and Maddaloni and Peydró (2009).
1
Electronic copy available at: http://ssrn.com/abstract=1561171
market volatility (uncertainty) and a residual, the so-called variance premium (see, for
example, Carr and Wu (2009)), that reflects risk aversion and other non-linear pricing
effects, perhaps even Knightian uncertainty. Establishing which component drives the
strong comovements between the monetary policy stance and the VIX is therefore
particularly important.
Third, analyzing the relationship between monetary policy and the VIX and its
components may help clarify the relationship between monetary policy and the stock
market, explored in a large number of empirical papers (Thorbecke (1997), Rigobon and
Sack (2004), Bernanke and Kuttner (2005)). The extant studies all find that expansionary
(contractionary) monetary policy affects the stock market positively (negatively).
Interestingly, Bernanke and Kuttner (2005) ascribe the bulk of the effect to easier
monetary policy lowering risk premiums, reflecting both a reduction in economic and
financial volatility and an increase in the capacity of financial investors to bear risk. By
using the VIX and its two components, we test the effect of monetary policy on stock
market risk, but also provide more precise information on the exact channel.
This article characterizes the dynamic links between risk aversion, economic
uncertainty and monetary policy in a simple vector-autoregressive (VAR) system. Such
analysis faces a number of difficulties. First, because risk aversion and the stance of
monetary policy are jointly endogenous variables and display strong contemporaneous
correlation (see Figure 1), a structural interpretation of the dynamic effects requires
identifying restrictions. Such structural identification is of considerable interest, as there
may be causal relationships in either direction. Monetary policy may indeed affect asset
prices through its effect on risk aversion, as suggested by the literature on monetary
policy news and the stock market, but monetary policy makers may also react to a
nervous and uncertain market place by loosening monetary policy. In the literature on the
relationship between the stock market and monetary policy, another paper by Rigobon
and Sack (2003) finds that the Federal Reserve does systematically respond to stock
prices, for example. 3
3
The two papers by Rigobon and Sack (2003, 2004) use an identification scheme based on the
heteroskedasticity of stock market returns. Given that we view economic uncertainty as an important
endogenous variable in its own right with links to the real economy and risk premiums, we cannot use such
an identification scheme.
2
Electronic copy available at: http://ssrn.com/abstract=1561171
Second, the relationship between risk aversion and monetary policy may also reflect
the joint response to an omitted variable, with business cycle variation being a prime
candidate. Recessions may be associated with high risk aversion (see Campbell and
Cochrane (1999) for a model generating counter-cyclical risk aversion) and at the same
time lead to lax monetary policy. Third, there are a number of measurement problems.
Disagreement on how to measure risk aversion in financial markets is rife (see Baker and
Wurgler (2007) and Coudert and Gex (2008) for surveys of indicators of investor
sentiment in general and risk aversion in particular) and measuring the monetary policy
stance is the subject of a large literature (see, for example, Bernanke and Mihov (1998a)).
In addition, measuring policy shocks correctly is difficult. Models featuring time-varying
risk aversion and/or uncertainty, such as Bekaert, Engstrom and Xing (2009), imply an
equilibrium contemporaneous link between interest rates and risk aversion and
uncertainty, through precautionary savings effects for example. Such relation should not
be associated with a policy shock.
To overcome these difficulties, we strive to characterize the data as fully as possible.
In Section II, we start with an analysis of the VIX and a measure of the monetary policy
stance (the real interest rate) in a bivariate VAR. We show the dynamic effects of the
VIX on monetary policy and vice versa, under various identification schemes. We also
show robustness to a number of alternative measurements of the monetary policy stance.
In Section III, we expand the VAR to account for the most obvious omitted variable, a
business cycle indicator, and also split the VIX into a pure volatility component and a
residual, which should be more closely associated with risk aversion. In the final section,
we empirically examine various channels through which monetary policy may affect risk
aversion and private sector risk-taking behavior, as suggested by recent research.
Specifically, we consider the effects through the balance sheet of financial intermediaries
(as proxied by repo growth and the growth rates of broad money aggregates) and through
the expansion of credit (using the growth of credit and credit-to-GDP ratio).
Our main findings are as follows. A lax monetary policy decreases risk aversion in
the medium run, while uncertainty (stock market volatility) appears to be unaffected by
monetary policy. On the other hand, periods of high uncertainty are followed by a looser
monetary policy stance. These effects are persistent. Moreover, they are robust to
3
controlling for business cycle movements, as well as to using alternative measures of
monetary policy. The effect of monetary policy on risk aversion is independent and does
not necessarily run through repo or credit growth.
II. The VIX and Monetary Policy
We begin our analysis with a bivariate VAR on a measure of monetary policy and a
measure of risk aversion, using monthly data for the United States from January 1990 to
July 2007. Table 1 lists the variables, their labels and description. Note that we exclude
recent data on the crisis, which presents special challenges.
To measure the monetary policy stance, we use the real interest rate, i.e., the Fed
funds end-of-the-month target rate minus the CPI inflation rate. We then consider
alternative measures of the monetary policy stance for robustness: Taylor rule deviations,
the nominal Fed funds rate, and the growth of the monetary aggregate M1.
To measure risk aversion, we use end-of-month VIX levels. The VIX represents the
implied volatility of a hypothetical at-the-money option on the S&P500 index with a
horizon of 30 calendar days (22 trading days). Since 2004, this volatility estimate is based
on a weighted average of European-style S&P500 options that straddle a 30-day maturity
and cover a wide range of strikes (see CBOE (2004) for more details). The VIX is often
viewed as a measure of risk aversion in the market place, although it obviously also
reflects stock market uncertainty. We decompose the VIX information in Section III.
We collect these two variables in the vector Zt = [mpt, rat]' where mpt is a measure of
monetary policy stance and risk aversion rat is represented by the logarithm of the
implied volatility (LVIX). Without loss of generality, we ignore constants. Consider the
following structural VAR:
A Zt = Φ Zt-1 + εt
(1)
where A is a 2x2 full-rank matrix and E[εt εt'] = I. Of main interest are the dynamic
responses to the structural shocks εt.
Of course, we start by estimating the reduced-form VAR. We re-write (1) as follows:
Zt = B Zt-1 + C εt
(2)
where B denotes A-1 Φ and C denotes A-1. Moreover, let us define Σ to be the variancecovariance matrix of the reduced-form residuals, i.e., Σ = E[(C εt) (C εt)'] = C C'.
4
The first-order VAR in Equations (1) and (2) is useful to illustrate the identification
problem: Equation (2) yields 7 coefficients in the matrices B and Σ, but Equation (1) has
8 unknowns. We later use formal selection criteria to select the correct order of the VAR.
In general, for a VAR of order k with N variables, we have (k+1)N2 parameters to
identify and we can estimate kN2 + N(N+1)/2 parameters. Hence, we need N(N-1)/2
restrictions to identify the system.
Reduced-form Evidence
Before we explore structural identification, Table 2 reports some reduced-form VAR
statistics. Panel A produces three lag-selection criteria: Akaike (AIC), Hannan-Quinn
(HQIC) and Schwarz (SBIC). While the Schwarz and HQIC criteria select relatively
parsimonious VARs, the AIC criterion selects a VAR with 14 lags. We focus the
remainder of the analysis in this section on the 14 lag VAR, for reasons detailed below.
Panel B reports Granger causality tests. We find strong Granger causality in either
direction, i.e., the monetary policy stance predicts risk aversion and risk aversion predicts
the monetary policy stance. 4
Finally, Panel C reports some specification tests on the residuals of the VAR. These
tests (see Johansen (1995)) test for autocorrelation in the residuals of the VAR at lag j
(j=1,2,3). The VAR with 14 lags clearly eliminates all serial correlation in the residuals.
We couch our main results in the form of impulse-response functions (IRFs
henceforth), estimated in the usual way. We compute 90% bootstrapped confidence
intervals based on 1000 replications, and focus our discussion on significant responses.
Figure 2 shows orthogonalized “reduced-form” IRFs, i.e., the impulse responses
generated by the shock of either variable, using a Cholesky decomposition of the estimate
of the variance-covariance matrix. We order the real interest rate first and the VIX second
to capture the fact that the VIX, a stock market based variable, responds instantly to the
monetary policy shocks, while the monetary policy stance is relatively more slowmoving. The response at lag zero represents the off-diagonal element in the Cholesky
factor.
4
In the 14 lag VAR, reporting the feedback coefficients is not very informative. In a first-order VAR,
contractionary monetary policy predicts higher risk aversion next period whereas higher risk aversion
predicts laxer monetary policy next month. Both coefficients are significant at the 5% level.
5
A one standard deviation negative shock to the real rate (equivalent to 25.60 basis
points), after an initial increase, lowers LVIX by 0.0209 after 15 months. The impact
reaches a maximum of 0.0295 after 32 months. The effect remains (borderline)
significant up and till lag 52. A one standard deviation positive shock to LVIX
(equivalent to 0.1341) leads to an 11.23 basis points decrease in the real interest rate after
23 months. The decrease reaches a maximum level of 17.06 basis points after 41 months.
Hence, apart from a somewhat puzzling significant negative short-run response of the
VIX to real interest rates, our evidence reveals that lax monetary policy can indeed lower
the VIX (and, if the VIX represents risk aversion, increase risk appetite) in the medium
run (after one year), and that monetary policy also reacts to periods of high VIX levels by
relaxing monetary policy, with a somewhat longer lag.
Structural Evidence
To obtain structural identification, we generally investigate three types of restrictions:
exclusion restrictions on contemporaneous responses (setting coefficients in A to zero),
long-run restrictions and exclusion restrictions on the feedback matrix Φ. In the setting of
the two-variable VAR, the exclusion restriction that monetary policy does not
contemporaneously react to risk aversion ( a12 = 0) is identical to restriction implied by
the Cholesky decomposition studied above. Our long-run restriction is inspired by the
literature on long-run money neutrality: money should not have a long run effect on real
variables. Bernanke and Mihov (1998b) and King and Watson (1992) marshal empirical
evidence in favor of money neutrality using data on money (growth) and output (growth).
Following Blanchard and Quah (1989), the model with a long-run restriction (LR)
involves a long-run response matrix, denoted by D:
D  (I - B)-1 C
(3)
It follows that D D' = (I - B)-1 C C' [(I - B)-1]' = (I - B)-1 Σ [(I - B)-1]'. Hence, using the
estimates of B and Σ from the reduced-form VAR, we obtain D, and thus A-1 = C. 5 Our
assumption of long-run money neutrality in the two-variable system implies that d21 = 0:
D=
mp d11
ra  0
d12 
d 22 
(4)
5
To facilitate interpretation of the impulse responses, we adopt a sign normalization requiring that the
diagonal elements of A-1 be positive.
6
This restriction is much stronger than the standard view on money neutrality, which
holds that permanent monetary policy shocks do no have a long-run effect on (real)
output. Therefore, money neutrality involves a restriction, analogous to the restriction
embedded in Equation (4), in a VAR on money growth and output growth. Because the
VAR variables are stationary, our framework already implies that monetary policy does
not have a long run effect on risk aversion. The restriction in (4) represents “superneutrality”: the total effect of monetary policy on risk aversion is restricted to be zero. In
the more elaborate VARs of Section III, such long-run restrictions will be reserved to the
relationship between monetary policy and business cycle variables.
Recently, the use of long-run restrictions to identify VARs has come under attack
(see, for example, Chari, Kehoe and McGrattan (2008)). However, Christiano,
Eichenbaum and Vigfusson (2008) show that many of the problems can be overcome by
using long-run information (rather than a parsimonious VAR) to identify the long-run
restrictions. Although they advocate using a non-parametrically estimated spectral
density matrix, our VAR with 14 lags, the lag-length selected by the Akaike criterion,
effectively uses long-run information to identify the restrictions.
As a third type of restriction, we restrict the feedback matrix (the first-order lagged or
“short-run” response), imposing that monetary policy does not have a short-run effect on
risk aversion or, equivalently,  21  0 : 6
Φ=
mp 11 12 
ra  0  22 
(5)
This exogeneity assumption on risk aversion is consistent with external habit models,
such as Campbell and Cochrane (1999) and Bekaert, Engstrom and Xing (2009), where
the state variable driving risk aversion follows a univariate autoregressive process, but its
shocks may be correlated with other state variables.
All three identification schemes satisfy necessary and sufficient conditions for global
identification of structural vector autoregressive systems (see Rubio-Ramírez, Waggoner
and Zha (2009)). 7
6
Whatever the order of the VAR, the feedback restrictions are always imposed on the first-lag matrix.
In this paper, we only consider systems that are globally identified. This consideration will restrict the set
of specifications we report in Section III and in the Appendix.
7
7
Figure 3 shows the impulses-response functions under the last two structural
restrictions. In the model with the long-run restriction, a one standard deviation negative
shock to the real rate (equivalent to 16.02 basis points), after an initial increase, lowers
LVIX by 0.0141 after 36 months. The maximum impact is a decrease by 0.0221 after 57
months. A one standard deviation positive shock to LVIX (equivalent to 0.0647) first
increases the real rate by a maximum of 28.86 basis points after 11 months and then leads
to a decrease in the real rate by a maximum of 11.06 basis points after 58 months.
In the model with the short-run restriction, a one standard deviation negative shock to
the real rate (equivalent to 26.51 basis points) lowers LVIX by 0.0221 after 16 months.
The maximum impact is 0.0312 after 32 months. A one standard deviation positive shock
to LVIX (equivalent to 0.1402) leads to a 12.90 basis points decrease in the real rate after
23 months. The maximum impact is 17.58 basis points after 31 months.
Consequently, the effects of monetary policy on risk aversion, uncovered in the
reduced-form analysis, are preserved under reasonable structural restrictions. A
somewhat puzzling short-lived negative effect of interest rates on risk aversion remains
robust as well. In the opposite direction, the real interest rate decreases following a
positive shock to risk aversion in the model with the short-run restriction. However, in
the model with the long-run restriction, this decrease only becomes significant after more
than 50 lags and the initial effect is positive.
Robustness
We first consider the robustness of our results with respect to three alternative
measures of monetary policy stance in Table 3. The first measure we consider is Taylor
rule residuals, i.e., the difference between the nominal Fed funds rate and the Taylor rule
rate (TR rate). The TR rate is estimated as in Taylor (1993):
TRt = Inft + NatRatet + 0.5*(Inft - TargInf) + 0.5*OGt
(6)
where Inf is the annual inflation rate, NatRate is the “natural” real Fed funds rate
(consistent with full employment), which Taylor assumed to be 2%, TargInf is a target
inflation rate, also assumed to be 2%, and OG (output gap) is the percentage deviation of
real GDP from potential GDP. We assume that the growth of potential GDP is 3% per
year. We also consider other measures of the monetary policy stance, namely the nominal
8
Fed funds rate instead of the real rate, and the growth rate of the monetary aggregate M1,
which is commonly assumed to be under tight control of the central bank.
The results confirm that a looser monetary policy stance (lower Fed funds rate
interest rate relative to the Taylor rule rate, lower Fed funds rate, or higher M1 growth)
leads to lower implied volatility. Specifically, a one standard deviation negative shock to
the Taylor rule deviation (equivalent to 8.86 basis points) lowers LVIX by a maximum of
0.0413 (equivalent to 52.68% of its standard deviation) after 46 months in the structural
model with the long-run restriction. A one standard deviation negative shock to the
Taylor rule deviations (equivalent to 33.59 basis points) lowers LVIX by a maximum of
0.0864 (equivalent to 27.04% of its standard deviation) after 32 months in the structural
model with the short-run restriction.
The effects for the Fed funds rate are similar. A one standard deviation negative
shock to the Fed funds rate (equivalent to 13.76 basis points) lowers LVIX by a
maximum of 0.0121 (equivalent to 9.47% of its standard deviation) after 25 months in the
structural model with the long-run restriction. A one standard deviation negative shock to
the Fed funds rate (equivalent to 16.41 basis points) lowers LVIX by a maximum of
0.0215 (equivalent to 14.25% of its standard deviation) after 20 months in the structural
model with the short-run restriction.
A one standard deviation positive shock to the M1 growth (equivalent to 0.4489
percentage points) lowers LVIX by a maximum of 0.0103 (equivalent to 9.56% of its
standard deviation) after 23 months in the structural model with the long-run restriction.
A one standard deviation positive shock to the M1 growth (equivalent to 0.0661
percentage points) lowers LVIX by a maximum of 0.0266 (equivalent to 153.76% of its
standard deviation) after 16 months in the structural model with the short-run restriction.
So, we find a robust causal reaction of the VIX to all four monetary policy measures.
The negative (medium-term) effect risk aversion has on the monetary policy stance is
mostly preserved when imposing the Cholesky factorization or the short-run restriction,
but it does not appear in the model with the long-run restriction when alternative
measures of the monetary policy stance are used. The response of monetary policy to
implied volatility may just reflect a reaction of both variables to economic conditions, a
hypothesis we formally check in Section III.
9
Next, we consider the robustness of our results to the lag-length specification in the
VAR. The 14-lag VAR may be over-parameterized; consequently, we reduce the lag
length to 9 lags. By reducing the lag length to 9 lags, we obtain a saturation ratio of
slightly over 10. The saturation ratio is the total number of observations divided by total
number of parameters estimated. Second, we estimate a VAR with 1 lag, as selected by
the Schwarz criterion (SBIC). In both instances, our results (not reported) are similar to
the results reported in Table 3.
III. Risk, Uncertainty and Monetary Policy
In this section, we analyze a four-variable VAR. First, we add a business cycle indicator.
It is conceivable that the intriguing links between the VIX and monetary policy simply
reflect monetary policy and implied volatility jointly reacting to business cycle
conditions. For example, news indicating weaker than expected growth in the economy
may make a cut in the Fed funds target rate more likely, but at the same time cause
people to be effectively more risk averse, for example because a larger number of
households feel more constrained in their consumption relative to “habit,” or because
people fear a more uncertain future. To analyze business cycle effects, we use the log of
jobless claims as a recession indicator, bct.
Second, we split the VIX into a measure of stock market or economic uncertainty, uct,
and a residual that should be more closely associated with risk aversion, rat. While we
used the VIX index as a measure of risk aversion, it is also an index of stock market
volatility (economic uncertainty). The recent finance literature emphasizes its two
different components, true economic uncertainty (the physical conditional variance of the
stock market) and the remainder, which is called the variance premium (see, e.g., Carr
and Wu (2009)) and usually computed as the difference between the squared VIX and an
estimate of the conditional variance. 8 Recent finance models attribute this difference
either to non-Gaussian components in fundamentals and (stochastic) risk aversion (see,
for instance, Bekaert and Engstrom (2009), Bollerslev, Tauchen and Zhou (2009),
Drechsler and Yaron (2009)) or Knightian uncertainty (see Drechsler (2009)). The fact
8
In the technical finance literature, the variance premium is actually the negative of the construct that we
use. However, increased risk aversion makes the variance premium more negative, and we want our
variable to act as a risk aversion indicator.
10
that the variance premium is nearly always positive suggests that these non-linear
features of the data are important. In the context of an external habit model, Bekaert,
Engstrom and Xing (2009) recently show how “risk aversion” and “economic
uncertainty” may have different effects on asset prices. These differences may be
important to acknowledge in monetary policy transmission.
To decompose the VIX index into its two components, we borrow a measure of the
conditional variance of stock returns from Bekaert and Engstrom (2009), who project
monthly realized variances (computed using squared 5-minute returns) on a set of
instruments. We call the logarithm of this variance estimate “uncertainty” (uct). The
logarithm of the difference between the squared VIX and this conditional variance is our
improved risk aversion measure (rat), although it may also capture Knightian uncertainty
and other nonlinearities. Appendix 1 sets out a one-period discrete state economy to
provide intuition on the relationship between volatility, the VIX and risk aversion. It
shows how the variance premium monotonically increases in the level of risk aversion.
Consequently, we now analyze a four-variable system with Zt = [mpt, bct, uct, rat]'.
This may provide important inputs to a rapidly growing macroeconomic literature on
news-driven business cycles. Beaudry and Portier (2006), for example, present empirical
evidence suggesting that business cycle fluctuations may be driven to a large extent by
changes in stock market expectations, which anticipate total factor productivity
movements. Bloom (2009) also shows that “economic uncertainty” has real effects, in
particular it generates a sharp drop in employment and output, which rebounds in the
medium term, and a mild long-run overshoot. He explains these facts in the context of a
production model where uncertainty increases the region of inaction in hiring and
investment decisions of firms facing non-convex adjustment costs. In his empirical work,
Bloom uses the VIX index to create an index of “exogenous” volatility shocks. However,
as the VIX reflects both uncertainty and risk aversion, it is conceivable that it is the risk
aversion component of the VIX index that generates the real effects, not the economic
uncertainty component. Moreover, these shocks may be simply correlated with business
cycles, as predicted by external habit models, for example. 9 In a recent Economist article,
9
To be fair, Bloom (2009) attempts to identify exogenous shocks to the VIX, which are less likely to be of
a cyclical nature.
11
Blanchard (2009) describes the VIX index as an indicator of Knightian uncertainty,
arguing that such uncertainty may prolong the current crisis. In both cases, the
implication is that monetary policy may want to respond strongly to uncertainty shocks,
in Bloom’s case to economic uncertainty shocks, in Blanchard’s case to what we call risk
aversion shocks.
Reduced-form Evidence
Table 4 reports some basic VAR statistics. The selection criteria, reported in Panel A,
now select much shorter VAR orders, with the Akaike criterion only selecting a thirdorder VAR. When we estimate that VAR, there is some evidence of remaining serial
correlation at the second lag (see Panel C). Panel B reports Granger causality tests for the
three-lag VAR. We continue to find strong overall Granger causality in the risk aversion
and real interest rate equations. However, while for risk aversion the strong predictive
power appears to be still driven by monetary policy, risk aversion no longer significantly
predicts the real rate. Uncertainty and jobless claims do predict the monetary policy
stance. Granger causality is not significantly present in either the jobless rate equation or
the uncertainty equation. While the p-values for the various variables and the overall pvalues are relatively low in the jobless equation (with risk aversion predicting or
anticipating jobless rates significantly at the 5% level), uncertainty is really only related
to past uncertainty.
Figure 4 reports the orthogonalized reduced-form impulse responses for the fourvariable VAR. Motivated by the Granger causality results, we use a Cholesky ordering
with uncertainty ordered first, jobless claims second, followed by the real interest rate
and risk aversion. 10 Ordering jobless claims second is consistent with the exclusion
restrictions used in Christiano, Eichenbaum and Evans (2005), where the business cycle
variables also respond sluggishly to interest rates. A one standard deviation negative
shock to the real rate (equivalent to 31.30 basis points) lowers risk aversion by 0.0441
after 3 months. The impact reaches a maximum of 0.0686 after 11 months and remains
significant up and till lag 33. A one standard deviation negative shock to the real rate has
10
We have also experimented with alternative orderings, in particular ordering jobless claims first,
followed by uncertainty, real interest rate and risk aversion, and the results are similar.
12
no statistically significant impact on uncertainty. It immediately raises jobless claims (an
increase of 0.0063). The effect becomes statistically insignificant thereafter.
A one standard deviation positive shock to risk aversion (equivalent to 0.2827)
immediately increases jobless claims (an increase of 0.0068), an effect that remains
significant up and till lag 25. It has no statistically significant impact on the real rate or
uncertainty.
As for the impact of uncertainty shocks, a one standard deviation positive shock to
uncertainty (equivalent to 0.5728) lowers the real rate immediately (by 7.49 basis points).
The maximum impact of 14.71 basis points occurs after 5 months, and the impact
remains significant up and till lag 15. A one standard deviation positive shock to
uncertainty increases risk aversion significantly in the short-run with a maximum impact
of 0.1507 in the initial period. It has no statistically significant impact on jobless claims.
A one standard deviation positive shock to jobless claims (equivalent to 0.0557)
lowers the real rate immediately (by 6.07 basis points). The impact reaches a maximum
of 20.47 basis points after 13 months, and remains significant up and till lag 31. A one
standard deviation positive shock to jobless claims lowers risk aversion in the medium
term (between lags 10 and 41) with the maximum impact of 0.0654 at lag 24. It has no
statistically significant impact on uncertainty.
Reduced-form impulse responses indicate that the real interest rate responds primarily
to uncertainty shocks, rather than risk aversion shocks. On the other hand, real rate
shocks have a persistent effect on risk aversion while they do not affect uncertainty.
Structural Evidence
To identify a four-variable system, we need 6 restrictions on the VAR. Our strategy is
to verify a number of alternative identification schemes, and see whether the structural
results are robust.
One set of restrictions combines five contemporaneous restrictions (also imposed
under the Cholesky ordering above) with the assumption that the effects of monetary
policy on the business cycle cancel out over time, i.e., a form of long-run money
neutrality. 11 That is, we replace the contemporaneous exclusion restriction regarding the
11
Because we view jobless claims as a stationary variable, this assumption is different and stronger than
standard applications of long-run money neutrality. For robustness, we use the log-difference of industrial
13
effect of monetary policy on the business cycle by the analogous long-run restriction.
Consequently, the restrictions correspond to the following contemporaneous matrix A
and long-run matrix D:
a11
0

0

uc  0
a12
a 22
a32
0
a13
a 23
a33
0
a14 
a 24 
a34 

a 44 
(7)
 d11
d
 21
 d 31

d 41
d12
d 22
d13
d 23
0
d 33
(8)
d 42
d 43
d14 
d 24 
d 34 

d 44 
ra
mp
A=
bc
ra
mp
D=
bc
uc
This system satisfies necessary and sufficient conditions for global identification of
structural vector autoregressive systems (see Rubio-Ramírez et al. (2009)).
An alternative identification scheme combines two long-run money neutrality
restrictions with four restrictions on the first-order lagged response (short-run
restrictions) as follows:
mp
bc
Φ=
ra
uc
11 12

 21  22
31 0

0
0
 d11
0

d 31

uc  0
mp
bc
D=
ra
13 14 
 23  24 
33 34 

0  44 
(9)
d14 
d 24 
d 34 

d 44 
(10)
d12
d13
d 22
d 23
d 32
d 42
d 33
d 43
We assume that uncertainty is not affected by the other variables in the short-run and that
the business cycle does not have a short-run effect on risk aversion. This choice is
motivated by the Granger causality results as well as the cross-correlations between
variables, suggesting both risk aversion and uncertainty are relatively “exogenous”
variables with respect to the business cycle. Of course, a number of theories (Sharpe
production and the log-difference of hours worked as alternative business cycle variables, in which case the
restriction indeed implies that monetary policy has no long-run effect on the level of industrial output and
hours worked (see p. 17).
14
(1990), Campbell and Cochrane (1999)) suggest that risk aversion moves countercyclically. While such dynamics can still be accommodated in the VAR through
contemporaneous responses and links in the higher order feedback matrices, we relax the
risk aversion – business cycle feedback restriction in a number of alternative
identification schemes in the Appendix. We keep the long-run restriction for the effects
of monetary policy on the business cycle and additionally impose that the effects of
monetary policy on uncertainty sum to zero. This implies that whatever short-run effect
monetary policy may have on uncertainty, it must eventually be reversed and the total
effect must be zero. In the Appendix, we consider an analogous “neutrality” assumption
for risk aversion. The system in Equations (9) and (10) also satisfies necessary and
sufficient conditions for global identification
We report the resulting structural impulse-response functions in Figure 5. In a model
with contemporaneous/long-run restrictions, a one standard deviation negative shock to
the real rate (equivalent to 32.51 basis points) lowers risk aversion by 0.0413 after 3
months. The impact reaches a maximum of 0.0654 after 11 months and remains
significant up and till lag 36. In a model with short/long-run restrictions, a one standard
deviation negative shock to the real rate (equivalent to 9.08 basis points) lowers risk
aversion by 0.0491 after 6 months. The impact reaches a maximum of 0.0574 after 13
months and remains significant up and till lag 35. So, the effect of monetary policy on
risk aversion is consistent with the reduced-form results under both identification
schemes. The impact of a one standard deviation positive shock to risk aversion
(equivalent to 0.2889 in the model with contemporaneous/long-run restrictions and to
0.0366 in the model with short/long-run restrictions) on the real rate is mostly negative
but not statistically significant.
The impact of real rate shocks on uncertainty is not statistically significant. In the
other direction, in the model with contemporaneous/long-run restrictions, the real rate
decreases by 7.54 basis points in period 0 following a positive one standard deviation
shock to uncertainty (equivalent to 0.5856). The impact reaches a maximum of 14.18
basis points in period 5 and remains significant up and till period 16. In the model with
short/long-run restrictions, the real rate decreases by 6.50 basis points in period 0
following a positive one standard deviation shock to uncertainty (equivalent to 0.5826).
15
The impact reaches a maximum of 10.91 basis points in period 5, and remains significant
up and till period 6. Hence, we find a structural effect of uncertainty on the subsequent
monetary policy stance in the expected direction under both identification schemes.
As for interactions with the business cycle variable (Panels E - J), monetary policy
has no statistically significant effect on jobless claims. A one standard deviation positive
shock to jobless claims (equivalent to 0.0566 in the model with contemporaneous/longrun restrictions and to 0.0455 in the model with short/long-run restrictions) lowers the
real rate in the medium-term. Specifically, the real rate decreases by a maximum of 14.07
basis points after 19 months, with the impact remaining significant up and till lag 30. In a
model with short/long-run restrictions, the real rate decreases by 14.97 basis points after
12 months. The impact reaches a maximum of 15.51 basis points in period 15 and
remains significant up and till period 39. Hence, monetary policy reacts as expected to
business cycle fluctuations. Higher risk aversion increases jobless claims in the short to
medium run, but the effect is only statistically significant in the model with
contemporaneous/long-run restrictions (Panel G). In the other direction, higher jobless
claims increase risk aversion in the short-run but the effect is not statistically significant.
Such effect is consistent with habit-based theories of countercyclical risk aversion as in
Campbell and Cochrane (1999). There is no statistically significant interaction between
jobless claims and uncertainty in either direction. These results potentially shed new light
on the analysis in Bloom (2009), who found that uncertainty shocks generate significant
business cycle effects, using the VIX as a measure of uncertainty. Our results suggest that
the link between the VIX and the business cycle may well be driven by the risk aversion
rather than the uncertainty component of the VIX. 12
Finally, increases in uncertainty predict future increases in risk aversion under both
identification schemes, which is consistent with the results from the reduced-form
impulse responses (Panel L). Risk aversion has no significant effect on uncertainty (Panel
K).
12
Popescu and Smets (2009) analyze the business cycle behavior of measures of perceived uncertainty and
financial risk premia in Germany. They also find that positive financial risk aversion shocks have a large
and persistent negative impact on the economy and are more important in driving business cycles than
uncertainty shocks.
16
Robustness
Our results are largely robust to the specific identification scheme used. In Appendix
A, we report other identification schemes that we tried and we summarize the main
results. We restrict attention to cases that are globally identified. In all cases, we find that
monetary policy has a persistent positive effect on risk aversion, with the effect
concentrated around 14 to 35 months. The negative effect of uncertainty on the real
interest rate is always present but not always statistically significant under alternative
schemes. The effect of risk aversion on real interest rates is not robust across
specifications. The effect is also negative (and insignificant) in one alternative scheme,
but positive and significant in five other schemes. The latter schemes impose long run
money neutrality for risk aversion (see Table 8). That is, they force the total effect of
monetary policy on risk aversion to sum to zero, which is perhaps too strong an
assumption. Finally, we consistently find that uncertainty does not respond to monetary
policy shocks.
We also check robustness to lag-length. A VAR with 4 lags eliminates all serial
correlation in the residuals and still has a saturation ratio of 10 (see Section II, p. 10 for a
discussion of the saturation ratio). The results are unaltered. To check for robustness with
respect to our business cycle measure, jobless claims, we estimate the model using the
unemployment rate, the log-difference of hours worked and the log-difference of
industrial production as alternative business cycle variables. When using hours worked or
industrial production, the long-run money neutrality assumption has the standard
interpretation of monetary policy having no long-run effect on the business cycle
variable, measured in levels. The estimates again are consistent with our previous
findings.
We conclude that a lax monetary policy decreases risk aversion significantly, with the
effect being significant in the medium run, while the real interest rate tends to decrease in
response to high uncertainty. The latter effect’s statistical significance is less robust. So,
our previous results extend to the four-variable set-up, i.e., even controlling for business
cycle movements and extracting the risk aversion component out of the VIX. Uncertainty
(stock market volatility) appears to be unaffected by the other factors.
17
IV. Channels
We have unearthed some intriguing interactions between the component in the VIX index
not related to actual stock market volatility, and the stance of monetary policy. If
monetary policy indeed affects risk aversion, our results could be important in the current
debate about the origins of the 2007-2009 crisis. While pinpointing in detail how
monetary policy affects risk aversion is beyond the scope of the article, we use this
section to empirically analyze some potential channels, discussed in a number of recent
articles.
Adrian and Shin (2008) suggest that the link between monetary policy and asset
prices runs through the balance sheets of financial intermediaries and that repo growth
rates adequately proxy for the riskiness of balance sheets. Using US data, they find that
the growth of outstanding repos forecasts the difference between implied and realized
volatility and that rapid growth in repos is associated with loose monetary policy (defined
as the Fed funds rate).
To examine the Adrian-Shin channel in our structural framework, we use a fourvariable VAR as in the previous section but with repo growth replacing the “business
cycle” variable. First, we examine whether introduction of this variable eliminates the
effect of the real interest rate on risk aversion we uncovered in the previous section.
Table 6 summarizes the results. Lax monetary policy is still associated with lower risk
aversion after 3 to 5 months (depending on the specification used). This effect is
persistent. In the opposite direction, the responses are not statistically significant. In
Table 7, we investigate the interaction between the real rate and uncertainty, finding the
results to be very similar to those obtained in the previous section: the real rate decreases
following a positive shock to uncertainty.
Second, we analyze the direct link between repo growth and risk aversion. While
higher repo growth has a negative effect on risk aversion under both structural
specifications, the effect is not statistically significant. Hence, our VAR suggests that the
monetary policy – risk aversion link does not only run through repo growth.
Many commentators have noted a rather large build up of liquidity through money
growth prior to financial crises (see also Adalid and Detken (2007), Alessi and Detken
(2009)). We thus use the growth rates of a broad money aggregate as a “channel”
18
variable, replacing the business cycle variable in the four-variable VAR. In particular, we
consider the growth rate of M2 net of M1. This part of the money growth is arguably less
under control of a central bank and rather reflects activities of the financial sector. It may
also reflect savings behavior of individuals, who might flee into safe assets during crisis
times.
Using this set-up, we confirm our finding that lower real rates lead to lower risk
aversion in all specifications considered (see Table 6). We also confirm that positive
uncertainty shocks lower the real rate (Table 7). As for the impact of money growth on
risk aversion, we find that higher growth rate of (M2 - M1) has a positive impact on risk
aversion. This effect is significant between lags 5 and 23 in the model with
contemporaneous/long-run restrictions and at lag 4 in the model with short/long-run
restrictions. Estimating the same system with M1 growth replacing (M2 - M1) growth
reveals that higher M1 growth lowers risk aversion significantly between lags 7 and 38 in
the model with contemporaneous/long-run restrictions and between lags 10 to 39 in the
model with short/long-run restrictions. The interactions of risk aversion and uncertainty
with the real rate remain unchanged. In sum, once a broader monetary aggregate is
cleansed of M1 growth, the relation with risk aversion turns positive. This result is
surprising as we would expect high liquidity to be associated with low risk aversion. It is
of course conceivable that the finding is related to flights-to-safety effects in the sense
that risk-averse investors may flee to relatively safe assets that are incorporated in the M2
measure (e.g., money market and time deposits). However, if this is true, we should find
a structural link from risk aversion to M2-M1 and not vice versa.
According to Borio and Lowe (2002), medium-term swings in asset prices are
associated with a rapid credit expansion, Moreover, they stress that such financial
imbalances may build up in a low inflation environment and that in some cases it is
appropriate for monetary policy to respond to these imbalances. Consequently, they
suggest a link between credit growth and monetary policy. It is conceivable that periods
of high risk appetite coincide with periods of rapid credit expansion, suggesting a channel
for the effect of monetary policy on risk aversion.
To investigate the role of credit, we consider two separate four-variable VAR
systems, with (private) credit growth and the first-difference of the credit-to-GDP ratio
19
replacing the “business cycle” variable. The significant impact of monetary policy on risk
aversion is present again (see Table 6). Higher uncertainty decreases the real rate in all
specifications, with the effect being statistically insignificant for the credit-to-GDP ratio
in the model with short/long-run restrictions (see Table 7). We do not find statistically
significant effects of credit developments on risk aversion in the stock market.
While our results are robust to three different identification schemes, two of them rely
on a long-run money neutrality assumption that is less palatable for our channel variables
than it is for the business cycle variable, to which it was applied in the previous section.
We therefore examine one more alternative identification scheme, switching the channel
and uncertainty variables in the Cholesky ordering we use for the first identification
scheme. Again, the results are robust. In sum, considering channels through which
monetary policy may affect risk aversion does not eliminate the direct effect of the real
interest rate on risk appetite. Moreover, higher M1 growth has an independent,
persistently negative impact on risk aversion.
V. Conclusions
A number of recent studies point at a potential link between loose monetary policy and
excessive risk-taking in financial markets. Rajan (2006) conjectures that in times of
ample liquidity supplied by the central bank, investment managers have a tendency to
engage in risky, correlated investments. To earn excess returns in a low interest rate
environment, their investment strategies may entail risky, tail-risk sensitive and illiquid
securities (“search for yield”). Moreover, a tendency for herding behaviour emerges due
to the particular structure of managerial compensation contracts. Managers are evaluated
vis-à-vis their peers and by pursuing strategies similar to others, they can ensure that they
do not under perform. This “behavioral” channel of monetary policy transmission can
lead to the formation of asset prices bubbles and can threaten financial stability. Given
the dramatic crisis witnessed in 2007-2009, Rajan’s story sounds prophetic. Yet, there is
no empirical evidence on the links between risk aversion in financial markets and
monetary policy.
This article has attempted to provide a first characterization of the dynamic links
between risk, uncertainty and monetary policy, using a simple vector-autoregressive
20
framework. We find a robust structural interaction between implied volatility and
monetary policy. We decompose implied volatility into two components, risk aversion
and uncertainty, and find interactions between each of the components and monetary
policy to be rather different. Lax monetary policy increases risk appetite (decreases risk
aversion) in the future, with the effect lasting for more than two years and starting to be
significant after about six months. Uncertainty appears to be exogenous and does not
respond to monetary policy. On the other hand, high uncertainty leads to laxer monetary
policy in the future, with the effect lasting for over a year. These results are robust to
controlling for business cycle movements. Consequently, our VAR analysis provides a
clean interpretation of the stylized facts regarding the dynamic relations between the VIX
and the monetary policy stance depicted in Figure 1. The primary component driving the
co-movement between past monetary policy stance and current VIX levels (first column
of Figure 1) is risk aversion. The uncertainty component of the VIX lies behind the
negative relation in the opposite direction (second column of Figure 1).
We hope that our analysis will inspire further empirical work and research on the
exact theoretical links between monetary policy and risk-taking behavior in asset
markets. In particular, recent work in the consumption-based asset pricing literature
attempts to understand the structural sources of the VIX dynamics (see Bekaert and
Engstrom (2009), Bollerslev, Tauchen and Zhou (2008), Drechsler and Yaron (2009)).
Yet, none of these models incorporates monetary policy equations. In macroeconomics, a
number of articles have embedded term structure dynamics into the standard NewKeynesian workhorse model (Bekaert, Cho, Moreno (2010), Rudebusch and Wu (2008)),
but no models accommodate the dynamic interactions between monetary policy, risk
aversion and uncertainty, uncovered in this article. The effects missed are economically
important. In Figure 6, we show structural variance decompositions calculated from the
four-variable VAR for risk aversion and for monetary policy. Monetary policy shocks
account for approximately 25-30% of the variance of risk aversion (depending on the
identification scheme used) at horizons longer than 25 months. Stock market volatility,
our proxy for economic uncertainty, accounts for 10-18% of the variance of our monetary
policy variable, and is at least as important as business cycle variation at short horizons.
21
The policy implications of our work are also potentially very important. Fed chairman
Bernanke (see Bernanke (2002)) interprets his work on the effect of monetary policy on
the stock market (Bernanke and Kuttner (2005)) as suggesting that monetary policy
would not have a sufficiently strong effect on asset markets to pop a “bubble” (see also
Bernanke and Gertler (2001), Gilchrist and Leahy (2002), and Greenspan (2002)).
However, if monetary policy significantly affects risk appetite in asset markets, this
conclusion may not hold. If one channel is that lax monetary policy induces excess
leverage as in Adrian and Shin (2008), perhaps monetary policy is potent enough to weed
out financial excess. Conversely, in times of crisis and heightened risk aversion,
monetary policy can influence risk aversion in the market place, and therefore affect real
outcomes. Blanchard (2009) noted that the economy and financial markets had “nothing
to fear but fear itself”, suggesting a role for policy to reduce these fears. His conclusion
that markets were “fearful” was exactly inspired by unusually elevated VIX levels.
One disadvantage of our framework is that it does not really test the Rajan (2006) and
related stories. Current stories about the potential pernicious effects of lax monetary
policy give a prominent role to the length of the policy, and are explicitly asymmetric
(they are about policy being too lax, not too contractionary). Such features are really not
present in our linear VAR framework. We conduct one preliminary and informal test,
investigating whether the duration of a lax policy regime would play a role in the
dynamic interactions we observe. To that end, we construct a “duration-adjusted
monetary policy” (DUMP for short) variable which takes on the value of
log(1+dt+dt2)*TRdevt, where dt is the number of periods for which the deviation from the
Taylor rule, TRdev, is negative. If the deviation is positive, DUMP is given by the
TRdevt itself. This variable thus puts more weight on those periods in which monetary
policy was loose for a prolonged period of time. We estimate the four-variable system
composed of DUMP, jobless claims, uncertainty and risk aversion. We find that, in a
model with contemporaneous/long-run restrictions, a one standard deviation negative
shock to DUMP lowers risk aversion by 0.0497 after 3 months. The impact reaches a
maximum of 0.0711 after 9 months. In a model with short/long-run restrictions, a one
standard deviation negative shock to duration lowers risk aversion by a maximum of
0.0677 after 3 months.
22
In sum, monetary policy that is “too low for too long” decreases risk aversion in the
stock market. The impact is somewhat stronger and more immediate than the one we
found using unadjusted indicators of monetary policy stance. While suggestive, we plan
to investigate potential asymmetric and duration effects more formally in an explicitly
non-linear framework in the near future.
23
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26
Table 1: Description of variables
Name
Label
Description
Credit growth
CG
Month-on-month growth of business loans
Credit-to-GDP ratio
CGDP
Ratio of credit to GDP (intrapolated)
Fed funds rate
FED
Fed funds target rate
Hours worked
HW
Average weekly hours (private industries)
Implied volatility S&P500
LVIX
Log of (VIX / 12 )
Industrial production
IP
Industrial production index
Jobless claims
LJOB
Log of jobless claims
M1 money aggregate growth
M1
Month-on-month growth of M1
M2 net of M1 money growth
M2-M1
Month-on-month growth of (M2-M1)
Real interest rate
RERA
FED minus annual CPI inflation rate
Repo growth
GREPO
Monthly growth in repos outstanding
Risk aversion
RA
Log of (squared VIX / 12 minus exp(UC))
Taylor Rule deviations
TRULE
FED minus TaylorRuleRate (see p.7)
Uncertainty (conditional variance)
UC
Log of (conditional variance / 12)
Unemployment rate
URATE
Unempl. rate minus 3-year moving average
Notes: Monthly frequency, end-of-the-moth data (seasonally adjusted where applicable). Source: Thomson
Datastream; data on risk aversion and uncertainty are from Bekaert and Engstrom (2009).
27
Figure 1: Cross-correlogram LVIX RERA
Notes: The first column presents the (lagged) cross-correlogram between the log of the VIX (LVIX) and
past values of the real interest rate (RERA). The second column presents the (lead) cross-correlogram
between LVIX and future values of RERA. Dashed vertical lines indicate 95% confidence intervals for the
cross-correlation. The third column presents the cross-correlation values. The index i indicates the number
of months either lagged or led for the real interest rate variable.
28
Table 2: Bivariate VAR results (RERA, LVIX)
Panel A: Lag-length selection
lag
AIC
HQIC
SBIC
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-0.1170
-0.1383
-0.2496
-0.2686
-0.2776
-0.2691
-0.2525
-0.2318
-0.2469
-0.2298
-0.2534
-0.2453
-0.4196
-0.5056*
-0.4717
-0.0899
-0.0841
-0.1683*
-0.1603
-0.1421
-0.1066
-0.0629
-0.0151
-0.0031
0.0411
0.0445
0.0797
-0.0675
-0.1265
-0.0655
-0.0501*
-0.0045
-0.0488
-0.0010
0.0570
0.1323
0.2158
0.3034
0.3552
0.4392
0.4825
0.5575
0.4501
0.4310
0.5318
Panel B: Granger causality
Equation
Excluded
chi2
df
p-value
LVIX
RERA
RERA
LVIX
28.7290
48.2480
14
14
0.0110
0.0000
Panel C: Lagrange-multiplier test
lag
chi2
df
p-value
1
2
3
1.5554
3.0961
5.4070
4
4
4
0.8168
0.5419
0.2480
Notes: Bivariate VAR on the log of VIX (LVIX) and the real interest rate (RERA). Panel A presents laglength selection results based on three criteria: Akaike (AIC), Hannan-Quinn (HQIC) and Schwarz (SBIC).
The star indicates the lag chosen. Panel B presents Granger causality results for the model with 14 lags
(selected by Akaike). Panel C presents Lagrange-multiplier specification tests for the model with 14 lags
(selected by Akaike). The null hypothesis is that there is no autocorrelation at lag order j=1,2,3 and the
degrees of freedom are given by the square of the number of equations in the VAR, as the test examines the
null hypothesis that the residuals of lag j are not jointly significant in the VAR.
29
Figure 2: Orthogonalized reduced-form IRFs (RERA, LVIX)
Panel A: Impulse RERA, response LVIX
0.06
0.04
0.02
0
0
4
8
12
16
20
24
28
32
36
40
44
48
52
56
60
48
52
56
60
-0.02
-0.04
-0.06
Panel B: Impulse LVIX, response RERA
0.30
0.20
0.10
0.00
0
4
8
12
16
20
24
28
32
36
40
44
-0.10
-0.20
-0.30
-0.40
Notes: Medians (blue lines) and 90% confidence intervals (grey lines) of the estimated orthogonalized IRFs
for the model with 14 lags (selected by Akaike). Order of variables in the VAR is RERA LVIX. Panel A
presents the response of LVIX to a one standard deviation shock to RERA. Panel B presents the response
of RERA to a one standard deviation shock to LVIX.
30
Figure 3: Structural-form IRFs (RERA, LVIX)
Panel A: Impulse RERA, response LVIX
Long-run restriction
Short-run restriction
0.07
0.10
0.05
0.05
0.03
0.00
0.01
0
4
8
12 16 20 24 28 32 36 40 44 48 52 56 60
-0.01 0
-0.05
4
8
12 16 20 24 28 32 36 40 44 48 52 56 60
-0.03
-0.10
-0.05
-0.15
-0.07
Panel B: Impulse LVIX, response RERA
Long-run restriction
0.50
Short-run restriction
0.25
0.40
0.15
0.30
0.05
0.20
0.10
-0.05 0
0.00
-0.10
-0.20
-0.30
0
4
8 12 16 20 24 28 32 36 40 44 48 52 56 60
4
8 12 16 20 24 28 32 36 40 44 48 52 56 60
-0.15
-0.25
-0.35
Notes: Medians (blue lines) and 90% bootstrapped confidence intervals (grey lines) of the distribution of
the estimated structural impulse-response functions for the model with 14 lags (selected by Akaike), based
on 1000 replications. Panel A presents the response of LVIX to a one standard deviation shock to RERA.
Panel B presents the response of RERA to a one standard deviation shock to LVIX. Panels on the left
present results of the model with the long-run restriction, panels on the right present results of the model
with the short-run restriction.
31
Table 3: Robustness to monetary policy measures
MP instrument
Impulse MP, response LVIX
Impulse LVIX, response MP
sign
significant from-to (month)
sign
significant from-to (month)
- reduced-form
+
15 – 17, 21 – 52, 57

23 - 59
- structural LR
+
36 – 53, 55 - 58

55 - 59
- structural SR
+
16-17, 21-44, 46-48, 51-52

23 - 58
- reduced-form
+
7 - 10, 12, 14 - 43

23 - 25, 31, 40 - 52
- structural LR
+
31 – 58, 60

--
- structural SR
+
12, 14 – 53

22 - 25, 31 - 34, 38 - 59
- reduced-form
+
18 - 29

2 - 18
- structural LR
+
24 – 25, 27

--
- structural SR
+
17 - 34

4 - 15
- reduced-form

6 - 30
+
--
- structural LR

17 - 40
+
--
- structural SR

16 – 22
+
--
Real interest rate
Taylor rule
Fed funds Rate
M1 growth
Notes: Table 3 summarizes results for the real interest rate (RERA) and three alternative measures of the
monetary policy stance: Taylor rule deviations (Taylor rule), Fed funds rate, and growth of the monetary
aggregate M1. It lists for how many months impulse-response functions (from the reduced-form VAR, and
the structural VARs with long-run and short-run restrictions, respectively) were statistically significant
within the 90% confidence interval in the direction indicated in the column “sign”.
32
Table 4: Four-variable VAR results (RA, RERA, LJOB, UC)
Panel A: Lag-length selection
lag
AIC
HQIC
SBIC
1
2
3
4
5
6
7
8
9
10
0.0555
0.0643
-0.0216*
-0.0088
0.0131
0.1223
0.1942
0.2652
0.2559
0.3212
0.1639*
0.2810
0.3034
0.4246
0.5548
0.7723
0.9526
1.1319
1.2310
1.4046
0.3231*
0.5995
0.7812
1.0617
1.3511
1.7279
2.0674
2.4060
2.6643
2.9973
Panel B: Granger causality
Equation
Excluded
chi2
df
p-value
RA
RA
RA
RA
RERA
RERA
RERA
RERA
LJOB
LJOB
LJOB
LJOB
UC
UC
UC
UC
RERA
LJOB
UC
ALL
RA
LJOB
UC
ALL
RA
RERA
UC
ALL
RA
RERA
LJOB
ALL
14.4460
1.5888
5.7468
24.8780
4.4763
13.7200
9.2123
22.1440
9.1933
6.4407
5.5901
12.4980
0.2420
4.6045
0.9810
6.8800
3
3
3
9
3
3
3
9
3
3
3
9
3
3
3
9
0.0020
0.6620
0.1250
0.0030
0.2140
0.0030
0.0270
0.0080
0.0270
0.0920
0.1330
0.1870
0.9710
0.2030
0.8060
0.6500
Panel C: Lagrange-multiplier test
lag
chi2
df
p-value
1
2
3
22.0339
31.7639
21.4902
16
16
16
0.1421
0.0107
0.1604
Notes: Four-variable VAR on the log of risk aversion (RA), the real interest rate (RERA), the log of jobless
claims (LJOB) and the log of uncertainty (UC). Panel A presents lag-length selection results based on three
criteria: Akaike (AIC), Hannan-Quinn (HQIC) and Schwarz (SBIC). The star indicates the lag chosen.
Panel B presents Granger causality results for the model with 3 lags (selected by Akaike). Panel C presents
Lagrange-multiplier specification tests for the model with 3 lags (selected by Akaike). The null hypothesis
is that there is no autocorrelation at lag order j=1,2,3 and the degrees of freedom are given by the square of
the number of equations in the VAR, as the test examines the null hypothesis that the residuals of lag j are
not jointly significant in the VAR.
33
Figure 4: Orthogonalized reduced-form IRFs, UC LJOB RERA RA
Panel A: Impulse RERA
Response UC
Response RA
0.09
0.06
0.09
0.010
0.04
0.005
-0.01
0.03
0.00
0
6
12 18 24
30 36 42 48 54 60
-0.03
0
6
12 18 24 30 36 42 48 54 60
0.000
0
-0.06
-0.005
-0.11
-0.010
-0.16
-0.015
Panel B: Impulse RA
Response UC
Response RERA
0.10
6
12 18 24 30 36 42 48 54 60
Response LJOB
0.06
0.03
-0.04
Response LJOB
0.02
0.02
0
6
12
18
24
30
36
42
48
54 60
0.01
-0.02 0
-0.11
-0.06
-0.18
-0.10
6
12
18
24
30
36
42
48
54
60
0.00
0
6
30
36
42
48
54
60
0.012
-0.03 0
0.10
24
Response LJOB
0.03
0.16
18
-0.01
Panel C: Impulse UC
Response RERA
Response RA
12
6
12
18
24
30 36
42
48
54
60
0.004
-0.09
0.04
-0.15
-0.02 0
6
12
18
24
30
36
42
48
54
60
-0.08
0
-0.004
12 18
24 30 36
42 48 54 60
-0.21
-0.27
-0.012
Panel D: Impulse LJOB
Response UC
Response RA
6
Response RERA
0.15
0.03
-0.02 0
6
12
18
24
30
36
42
48
54
60
0.10
0.03
0.05
-0.06
-0.12
6
12
18
24
30
36
42
48
54
60
-0.15
0.00
0
-0.07
0
6
12
18
24
30
36
42
48
54
60
-0.05
-0.24
-0.10
-0.33
Notes: Medians (blue lines) and 90% confidence intervals (grey lines) of the estimated orthogonalized IRFs
for the model with 3 lags (selected by Akaike). Order of variables in the VAR is UC LJOB RERA RA.
Each panel presents the responses of three variables to a one standard deviation shock to the impulse
variable.
34
Figure 5: Structural-form IRFs for the 4-variable VAR (RERA, LJOB, UC, RA)
Panel A: Impulse RERA, response RA
Contemporaneous/long-run restrictions
Short/long-run restrictions
0.15
0.20
0.10
0.10
0.05
0.00
0.00
-0.10
0
0
4
8
12
16
20
24
28
32
36
40
44
48
52
56
60
-0.05
4
8
12 16 20 24 28 32 36 40 44 48 52 56 60
-0.20
Panel B: Impulse RA, response RERA
Contemporaneous/long-run restrictions
Short/long-run restrictions
0.15
0.20
0.05
0.00
0
0
4
8
4
8
12 16 20 24 28 32 36 40 44 48 52 56 60
12 16 20 24 28 32 36 40 44 48 52 56 60
-0.05
-0.20
-0.15
-0.40
Panel C: Impulse RERA, response UC
Contemporaneous/long-run restrictions
Short/long-run restrictions
0.10
0.10
0.00
0
0.00
0
4
8
12 16 20 24 28 32 36 40 44 48 52 56 60
4
8
12 16 20 24 28 32 36 40 44 48 52 56 60
-0.10
-0.10
-0.20
-0.20
-0.30
Panel D: Impulse UC, response RERA
Contemporaneous/long-run restrictions
0.05
Short/long-run restrictions
0.05
0
4
8
12 16 20 24 28 32 36 40 44 48 52 56 60
0
-0.05
-0.05
-0.15
-0.15
-0.25
-0.25
4
8
12 16 20 24 28 32 36 40 44 48 52 56 60
35
Panel E: Impulse RERA, response LJOB
Contemporaneous/long-run restrictions
Short/long-run restrictions
0.03
0.03
0.00
0.00
0
4
8
0
12 16 20 24 28 32 36 40 44 48 52 56 60
-0.03
-0.03
-0.05
-0.05
4
8
12 16 20 24 28 32 36 40 44 48 52 56 60
Panel F: Impulse LJOB, response RERA
Contemporaneous/long-run restrictions
Short/long-run restrictions
0.20
0.20
0.10
0.10
0.00
0.00
0
4
8
0
12 16 20 24 28 32 36 40 44 48 52 56 60
-0.10
-0.10
-0.20
-0.20
-0.30
-0.30
4
8
12
16
20
24
28
32
36
40
44
48
52
56
60
Panel G: Impulse RA, response LJOB
Contemporaneous/long-run restrictions
Short/long-run restrictions
0.03
0.03
0.02
0.02
0.01
0.01
0.00
0
0.00
0
4
8
12 16 20 24 28 32 36 40 44 48 52 56 60
-0.01
4
8
12 16 20 24 28 32 36 40 44 48 52 56 60
-0.01
-0.02
Panel H: Impulse LJOB, response RA
Contemporaneous/long-run restrictions
Short/long-run restrictions
0.20
0.10
0.05
0.10
0.00
0
4
8
12 16 20 24 28 32 36 40 44 48 52 56 60
0.00
0
-0.05
-0.10
4
8
12 16 20 24 28 32 36 40 44 48 52 56 60
-0.10
36
Panel I: Impulse UC, response LJOB
Contemporaneous/long-run restrictions
Short/long-run restrictions
0.02
0.02
0.01
0.01
0
4
8
12 16 20 24 28 32 36 40 44 48 52 56 60
0
-0.01
-0.01
-0.02
-0.02
4
8
12 16 20 24 28 32 36 40 44 48 52 56 60
Panel J: Impulse LJOB, response UC
Contemporaneous/long-run restrictions
Short/long-run restrictions
0.20
0.15
0.10
0.05
0
0.00
0
4
8
12 16 20 24 28 32 36 40 44 48 52 56 60
-0.10
4
8
12 16 20 24 28 32 36 40 44 48 52 56 60
-0.05
-0.15
Panel K: Impulse RA, response UC
Contemporaneous/long-run restrictions
Short/long-run restrictions
0.10
0.15
0.05
0.05
0.00
-0.05
0
4
8
0
4
8
12 16 20 24 28 32 36 40 44 48 52 56 60
12 16 20 24 28 32 36 40 44 48 52 56 60
-0.05
-0.15
-0.10
-0.25
Panel L: Impulse UC, response RA
Contemporaneous/long-run restrictions
Short/long-run restrictions
0.20
0.30
0.20
0.10
0.10
0.00
0
4
8
12 16 20 24 28 32 36 40 44 48 52 56 60
0.00
0
-0.10
4
8
12 16 20 24 28 32 36 40 44 48 52 56 60
-0.10
Medians (blue lines) and 90% bootstrapped confidence intervals (grey lines) of the distribution of the
estimated structural impulse-response functions for the model with 3 lags (selected by Akaike), based on
1000 replications. Panels on the left present results of the model with contemporaneous/long-run
restrictions, panels on the right present results of the model with short/long-run restrictions.
37
Table 6: Channels, real interest rate – risk aversion pair
Channel
Impulse RERA, response RA
Impulse RA, response RERA
sign
significant from-to (month)
sign
Significant from-to (month)
- reduced-form
+
3 – 39

--
- structural CLR
+
3 – 41

--
- structural SLR
+
5 – 20

--
- reduced-form
+
4 – 36

--
- structural CLR
+
6 – 36

--
- structural SLR
+
16 – 23

--
- reduced-form
+
3 – 39

--
- structural CLR
+
3 – 40

--
- structural SLR
+
8 – 15

--
- reduced-form
+
3 – 39

--
- structural CLR
+
3 – 42

--
- structural SLR
+
9 – 22

--
Repo growth
(M2-M1) growth
Credit growth
Credit/GDP
Notes: Table 6 summarizes results for the interaction between monetary policy (as represented by the real
interest rate) and risk aversion (RA) in the four-variable model with RERA, RA, UC and a Channel
variable. The Table lists the corresponding Channel variable (left column) and for how many months
impulse-response functions (from the reduced-form VAR, and the structural VAR with
contemporaneous/long-run and short/long-run restrictions, respectively) were statistically significant within
the 90% confidence interval in the direction indicated in the column “sign”.
38
Table 7: Channels, real interest rate - uncertainty pair
Channel
Impulse RERA, response UC
Impulse UC, response RERA
sign
significant from-to (month)
sign
Significant from-to (month)
- reduced-form
+
--

0 – 34
- structural CLR
+
--

0 – 41
- structural SLR
+
--

0 – 41
- reduced-form
+
--

0 – 19
- structural CLR
+
--

0 – 23
- structural SLR
+
--

4
- reduced-form
+
--

0 – 30
- structural CLR
+
--

0 – 37
- structural SLR
+
--

14 – 28
- reduced-form
+
--

0 – 31
- structural CLR
+
--

0 – 38
- structural SLR
+
--

--
Repo growth
(M2-M1) growth
Credit growth
Credit/GDP
Notes: Table 7 summarizes results for the interaction between monetary policy (as represented by the real
interest rate) and uncertainty (UC) in the four-variable model with RERA, RA, UC and a Channel variable.
The Table lists the corresponding Channel variable (left column) and for how many months impulseresponse functions (from the reduced-form VAR, and the structural VARs with contemporaneous/long-run
and short/long-run restrictions, respectively) were statistically significant within the 90% confidence
interval in the direction indicated in the column “sign”.
39
Figure 6: Structural Variance Decompositions
Panel A: Variance decomposition of RA
Contemporaneous/long-run restrictions
1
Short/long-run restrictions
1
0.8
RA
RERA
RA
RERA
LJOB
UC
LJOB
UC
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
1
5
9
13
17
21
25
29
33
37
41
45
49
53
57
1
5
9
13
17
21
25
29
33
37
41
45
49
53
57
Panel B: Variance decomposition of RERA
Contemporaneous/long-run restrictions
1
Short/long-run restrictions
1
0.8
RA
RERA
RA
RERA
LJOB
UC
LJOB
UC
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
1
5
9
13
17
21
25
29
33
37
41
45
49
53
57
1
5
9
13
17
21
25
29
33
37
41
45
49
53
57
Notes: Medians of the distribution of the estimated structural variance decompositions from the fourvariable model RA, RERA, LJOB and UC with 3 lags (selected by Akaike), based on 1000 replications.
Panels on the left present results of the model with the contemporaneous/long-run restrictions, panels on
the right present results of the model with the short/long-run restrictions.
40
Appendix 1: The VIX, Uncertainty and Risk Aversion
It may not be immediately obvious how the VIX is related to the actual (“physical”)
expected variance of stock returns and to risk aversion. In this appendix, we consider a
particularly simple discrete state economy to clarify these relations.
Imagine a stock return distribution with three different states xi , as follows:
Good state: x g    a with probability (1  p ) / 2 ,
Bad State : xb    a with probability (1  p ) / 2 ,
Crash state: xc  c with probability p,
where   0 , a  0 and c  0 are parameters to be determined. We set them to match
statistics in the data for the US stock market, the mean, the variance (standard deviation)
and the skewness, while fixing the crash probability at an empirically plausible number.
The mean is given by:
X 
1 p
1 p
xg 
xb  pc  (1  p )   pc .
2
2
The variance is given by:
V 2 
1 p
1 p
(  a  X ) 2 
(   a  X ) 2  p (c  X ) 2
2
2
and the skewness ( Sk ) by:
3
2
V Sk 
1 p
1 p
(  a  X ) 3 
(   a  X ) 3  p (c  X ) 3 .
2
2
Consider a one-period world in which the investor has a utility function over wealth
that is power utility and that in equilibrium she invests her entire wealth in the stock
market:
~
 (W0 R )1 
~
U (W )  E 
,
 1  
~
where R is the gross return on the stock market, W0 is initial wealth and  is the
coefficient of relative risk aversion.
The “pricing kernel” in this economy is given by the marginal utility, denoted by m ,
~
and is proportional to R . Hence, the stochastic part of the pricing kernel moves
41
inversely with the return on the stock market. When the stock market is down, marginal
utility is relatively high and vice versa.
The physical variance of the stock market is exogenous in this economy, and simply
given by V . In other words, the variance is computed using the actual probabilities. What
does the VIX measure? It measures the “risk-neutral” conditional variance. That is, it
uses the so-called “risk-neutral probabilities”, which are simply probabilities adjusted for
risk.
In particular, for a general state probability  j for state j , the risk neutral probability
is:

RN
j
j
mj
E m
j
R j 
E m
.
So, for a given  , we can easily compute the risk-neutral probabilities since R j  x j  1 .
Then, in general, the risk-neutral variance is given by:
K
2
VIX 2    RN
j (x j  X )
j 1
and the variance premium is:
K
VP  VIX 2  V   ( RN
  j )( x j  X ) 2
j
j 1
Because the risk-neutral probability puts more weight on the crash state and the crash
state induces plenty of additional variance, the variance premium is positive in this case.
The higher is risk aversion, the more weight the crash state will get, and the higher the
variance premium will be.
For our economy, the variance premium has a particularly simple form:
VP  ( gRN 
where  gRN 
1 p
1 p
)( x g  X ) 2  ( bRN 
)( xb  X ) 2  ( cRN  p )( xc  X ) 2
2
2
1  p (   a  1) 
1  p (   a  1) 
(c  1) 
,  bRN 
and  cRN  p
.
E m
E m
E m
2
2
Suppose the statistics to match are as follows: X  10 % on an annualized basis,
  15% on an annualized basis, Sk  1 and p  0.5% . The implied crash return is
then given by c  -25.0995 % . Table 8 below provides, for different values of the
42
coefficient of relative risk aversion  , the values for the VIX on an annualized basis in
percent (VIX), the log of the VIX on a monthly basis (LVIX), i.e., log(VIX / 12 ), the
annualized variance premium (VP), and our risk aversion proxy computed on a monthly
basis (RA), i.e., log(VIX 2 / 12   2 / 12 ).
Table 8: Numerical examples
Parameter
VIX
LVIX
VP
RA
 2
15.9891
1.5295
0.0031
0.9377
 4
17.6206
1.6266
0.0085
1.9634
 6
20.1642
1.7615
0.0182
2.7169
Notes: Table 8 presents values of the VIX on an annualized basis in percent (VIX), the log of the VIX on a
monthly basis (LVIX), the annualized variance premium (VP), and our proxy for risk aversion on a
monthly basis (RA) for different values of the coefficient of relative risk aversion γ, while keeping the
probability of the crash state fixed at p = 0.5%.
43
Appendix 2: Alternative Identification Schemes
To ensure that our main results in the four-variable model, [RERA, LJOB, UC, RA]', are
not sensitive to the identification scheme, we tried alternative schemes, while always
preserving a structure that satisfies necessary and sufficient conditions for global
identification. Table 9 below provides the order of variables in the system, together with
identifying restrictions on the left and summarizes main results on the right hand side.
We first list the contemporaneous/long-run model and the short/long-run model we
described in the main text. We then present other models we estimated.
Table 9: Estimated alternative identifying restrictions
Restrictions
ra
mp
bc
uc
mp
bc
ra
uc
mp
bc
uc
ra
 a 11
 0

 0

 0
 d 11

 d 21
d
 31
 d 41
a 14 
a 24 
a 34 

a 44 
d 14 

d 24 
d 34 

d 44 
a 12
a 13
a 22
a 32
a 23
a 33
0
0
d 12
d 22
d 13
d 23
0
d 42
d 33
d 43
 11

 21
 31

 0
 d 11

 0
d
 31
 0
 12
 22
 13  14 
 23  24 
 33  34 

0  44 
 11

 21
 31

 0
 d 11

 0
 0

 d 41
0
0
d 14 

d 24 
d 34 

d 44 
d 12
d 22
d 13
d 23
d 32
d 42
d 33
d 43
 12
 22
 13  14 
 23  24 
 33  34 

0  44 
0
0
d 12
d 22
d 13
d 23
d 32
d 42
d 33
d 43
d 14 

d 24 
d 34 

d 44 
Impulse - response
sign
significant from-to (month)
RERA - RA
+
3 – 36
RA - RERA

--
RERA - UC
+
--
UC – RERA

0 – 16
RERA - RA
+
6 – 35
RA - RERA

--
RERA - UC
+
--
UC – RERA

0–7
RERA - RA
+
6 – 37
RA - RERA

--
RERA - UC

2–3
UC – RERA

--
44
mp
bc
uc
ra
d11
0

0

0
d11
0

0

uc  0
mp
bc
ra
d12
d13
d 22
0
d 23
d 33
0
0
d12
d13
d 22
d 23
0
d 33
0
0
d12
d13
d 22
d 23
0
d 33
0
0
mp
ra
bc
uc
d11
0

0

0
mp
bc
ra
uc
 11

 21
 31

 41
 d 11

 0
 0

 0
 12
 22
 11

 21
 31

 41
 d 11

 0
 0

 0
mp
bc
uc
ra
0
0
d14 
d 24 
d 34 

d 44 
RERA - RA
+
13 – 39
RA - RERA
+
0 – 26
RERA - UC
+
--
UC – RERA

4 – 22
d14 
d 24 
d 34 

d 44 
RERA – RA
+
12 – 39
RA – RERA
+
0 – 27
RERA – UC
+
--
UC – RERA

--
d14 
d 24 
d 34 

d 44 
RERA - RA
+
14 – 42
RA - RERA
+
0 – 27
RERA - UC
+
--
UC – RERA

--
RERA - RA
+
14 – 41
RA - RERA
+
0 – 14
RERA - UC
+
--
UC – RERA

--
RERA – RA
+
14 – 41
RA – RERA
+
0 – 12
RERA – UC
+
--
UC – RERA

0 – 11
 13  14 
 23  24 
 33  34 

0  44 
d 14 

d 24 
d 34 

d 44 
d 12
d 22
d 13
d 23
d 32
d 42
d 33
d 43
 12
 22
 13  14 
 23  24 
 33  34 

0  44 
0
0
d 12
d 22
d 13
d 23
d 32
d 42
d 33
d 43
d 14 

d 24 
d 34 

d 44 
Notes: Table 9 summarizes the main results for the interaction between monetary policy (as represented by
the real interest rate, RERA), risk aversion (RA) and uncertainty (UC) in the four-variable model with
RERA, LJOB, UC and RA under various identification schemes. The Table lists the corresponding
identification assumptions for the structural VAR (left column) and for how many months impulseresponse functions were statistically significant within the 90% confidence interval in the direction
indicated in the column “sign”.
45